“Most of the various online invoice services come with an assortment of professional-looking templates.”

You run your own freelance photography business. One of your cousins is trying to start doing the same. He asks your advice for creating invoices for his clients. This is your advice for making the invoices look nice.

Most of the various online invoice services come with an assortment of professional-looking templates.

"Online" means on the internet. An "online service" is some kind of tool that you use on the Internet. For example, there are online services for managing your money, online dating services for finding a romantic partner, and so on.

A "template" is an example of something like a document, that you can make small changes to. For example, if you are going to make a slide show for a presentation, you might start off by choosing a template from a list in the software that you're using. The templates will each have a different design.